### Abstract

Original language | English |
---|---|

Pages (from-to) | 63-85 |

Journal | Journal of Symbolic Computation |

Volume | 51 |

Issue number | April 2013 |

Early online date | 4 Jul 2012 |

DOIs | |

Publication status | Published - 2013 |

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### Cite this

*Journal of Symbolic Computation*,

*51*(April 2013), 63-85. https://doi.org/10.1016/j.jsc.2012.03.007

}

*Journal of Symbolic Computation*, vol. 51, no. April 2013, pp. 63-85. https://doi.org/10.1016/j.jsc.2012.03.007

**Moment matrices, border bases and radical computation.** / Lasserre, J.B.; Laurent, M.; Mourrain, B.; Rostalski, P.; Trébuchet, P.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Moment matrices, border bases and radical computation

AU - Lasserre, J.B.

AU - Laurent, M.

AU - Mourrain, B.

AU - Rostalski, P.

AU - Trébuchet, P.

PY - 2013

Y1 - 2013

N2 - In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming its complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of Mourrain and Trébuchet (2005) are efficient and numerically stable for computing complex roots, algorithms based on moment matrices (Lasserre et al., 2008) allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.

AB - In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming its complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of Mourrain and Trébuchet (2005) are efficient and numerically stable for computing complex roots, algorithms based on moment matrices (Lasserre et al., 2008) allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.

U2 - 10.1016/j.jsc.2012.03.007

DO - 10.1016/j.jsc.2012.03.007

M3 - Article

VL - 51

SP - 63

EP - 85

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

IS - April 2013

ER -